How do I make one object hit another and make it move?
They are going to float away because they don't have a floor. Make a plane (size it to however big you want) and apply the rigid body to it, making it passive instead of active. Place the plane just below your objects.

After that, you should animate the blue object to hit the ball. Click "animated" in the rigid body settings for the blue object then animate it to hit the ball. You can animate the blue object to bump into the ball as many times as you want, but make sure that the ball stays on the plane. If you hit the ball too far, it will fall off the edge of the plane.